In the bowl of a stand mixer fitted with a paddle attachment, beat together sugar, eggs, and egg yolk until well combined.
In a large measuring cup, whisk together buttermilk, melted butter, and baking soda. Let sit for 3-5 minutes to activate baking soda. Mixture will slightly rise. Pour activated mixture into mixing bowl with sugar-egg mixture. Beat together until well incorporated.
In a large bowl, whisk together flour, baking powder, salt, and nutmeg. Add to mixing bowl. Mix on low speed until thorough there are no longer any dry streaks of flour. Scrape down sides of bowl as needed to ensure thorough mixing.
Transfer sticky dough to a medium bowl. Cover and chill in the fridge for at least an hour, up to 24 hours.
Attach candy thermometer to a large heavy bottomed pot (cast-iron or Dutch oven, preferred). Fill pot about 3-inches deep with fry oil or shortening. Set over low-medium heat to 375°F. Prepare dough while oil heats.
Transfer dough to a floured work surface. Dough will be rather sticky. Lightly flour the top of dough and pat or roll dough to a thickness slightly less than ½-inch. Using a floured 3 1/2-inch doughnut cutter (or round cookie cutters), stamp out doughnuts. Transfer doughnuts and doughnut holes to baking sheet. Gather scrape dough, knead together, and re-roll to stamp out more doughnuts. Brush off any excess flour from the cut doughnuts. Carefully place about 3-4 doughnuts at a time into hot oil. Do not overcrowd the pot. Cook for 1 minute on each side, until golden in color. Cook doughnut holes for about 45 seconds on each side. Drain cooked doughnuts on wire rack or paper towel lined baking sheet. Keep an eye on candy thermometer and adjust head accordingly to maintain a constant cooking temperature.
While doughnuts are still warm, toss in your choice of cinnamon sugar, powdered sugar, or sweet butter glaze. Enjoy immediately.